Problem 1321 (difficulty: 4/10)

Determine whether the following statements are true or false. Here \(\displaystyle f\) is a function from \(\displaystyle [a,b]\) to \(\displaystyle \R\).

(a) If \(\displaystyle f\) is monotonic, then \(\displaystyle f\) is of bounded variation.

(b) If \(\displaystyle f\) is continuous, then \(\displaystyle f\) is of bounded variation.

(c) If \(\displaystyle f\) is continuous and of bounded variation, then \(\displaystyle f\) is Lipschitz.

(d) If \(\displaystyle f\) is of bounded variation, then the interval \(\displaystyle [a,b]\) can be written as the union of countable many subintervals on each of which \(\displaystyle f\) is monotonic.

(e) If the \(\displaystyle \int_a^b\df\) Stieltjes-integral exists, then \(\displaystyle f\) is absolutely continuous.

(f) If \(\displaystyle f\) is absolutely continuous, then \(\displaystyle f\) is Riemann-integrable.

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